 Thank you. Great, so let's summarize a little bit where we were at last time. So make this little app before and split the separation, your binary here and the mass ratio here. So I'm gonna put zero here. I'm gonna put one here. Then I put another zero here. I'm gonna put infinity here. And so what I had told you before was that there's some regime over here where numerical relativity works quite well. And then there is this regime over here where a self-force methods or black hole perturbation theory, how it sometimes called works really well. So here you're expanding in q much less than one. Here you're just doing a three plus one decomposition, so three plus one. And then there's this regime somewhere around here where post-Newtonian methods work very well. And post-Newtonian methods, these are methods in which you expand. I'm gonna be very irritating right now because I'm gonna write g much less than one and one over c much less than one. Okay, so these are not dimensionless parameters and so the mathematicians in the audience are probably cringing, but I'm gonna try to make some of these statements a bit more precise and tell you exactly what we're going to be expanding about when we use post-Newtonian techniques. So remember these works provided the orbital separations are large relative to the characteristic length scales. And I told you it sort of does not work very well when you go to the extreme mass ratio limit. So I'm gonna be considering mostly what we call comparable mass binary. So binary is with roughly the same mass maybe off by a factor of 10 or so. And there's a long, long, long history to post-Newtonian methods. Like there is, so the first person to do a post-Newtonian expansion of general relativity was Einstein as far as I know. So this dates back to like the 20s or 30s. The theory has evolved significantly since then and there's some really nice references that you can follow or you can study if you want to learn more about this. Perhaps the most well-known one or the most used one by experts in the field is Blanchet's living reviews and relativity. Do you guys know what living reviews and relativity are? Yes? No, it's a website, it's published by Springer. It's living because when they convince you to write one of these, then you're supposed to be in charge of updating them. Like once every three or four or five years. Problem is that nobody does. So these are more like zombie reviews. A sort of state of half alive, half dead. Anywho, so that's a really good one and it's quite mathematically formal. I mean, people here I think would really appreciate Blanchet's treatment. It's quite rigorous, although perhaps a little bit compact in terms of how much it explains things. If this is the first time you're hearing about post-Newtonian, what Blanchet's review is perhaps not the place to start. So there's this book by Poisona and Will that came out recently called Gravity that you can also use and what's nice about this book is that it provides all of the glory details that you probably never wanted to learn but you will probably have to derive if you use that book up to formal one post-Newtonian or one PN order. Maybe they did it to PN, I don't know, I think just to one PN. Also there is a textbook by Majori who sort of introduces some of this concept well at an even lower level. So those are your references. At Montana, I teach four semesters of general relativity, one of which is entirely devoted to half of this book. So I'm gonna try to condense that in 75 minutes and we'll see how it goes, okay? So let's begin just to tie with Peter's first lecture with wave coordinates because these are coordinates are used quite a bit. So I'm gonna start by discussing the harmonic gauge condition and this will allow me to introduce some notation as well. So Peter told us that harmonic coordinates are coordinates so these are harmonic coordinates that satisfy this equation. So this box operator is supposed to be in the metric contracted twice with a covariant derivative acting on your coordinates and because this is a complete trace on these covariants Peter also wrote down that this was equal to one over root minus g d alpha of root minus g alpha beta d beta x mu equal to zero, okay? Okay, okay, great. So let me know and this should work in principle for any set of coordinates that you choose any x mu that satisfy that set of equations will be harmonic. However, it is customary in post Newtonian theory to employ Cartesian coordinates so x, y, z Cartesian coordinates or t x, y, z Cartesian coordinates for which this derivative acting on x mu turns into a chronicle delta. So now we have that this expression here turns into d beta acting on root minus g times g alpha beta has to be equal to zero which can also be written as the contraction of the metric with a Christoffel symbol, by the way. So what we do in post Newtonian theory is we introduce a quantity. So this thing in parenthesis here, this thing is gonna be called the gothic metric and I'm supposed to know how to write a calligraphic g but I don't know how to do that. So it's an eight. So eight is calligraphic g which stands for gothic metric and I have to put indices. So there you go. And so this thing is called the gothic metric. So your gauge condition becomes just d beta on the gothic g has to vanish, great. And so what we're gonna do in post Newtonian theory is since we are in this regime where fields are weak and velocities are small, the spacetime metric can be decomposed as the Minkowski metric plus a metric perturbation. So what we're gonna do now is we're gonna introduce the metric perturbation as h alpha beta is gonna be defined to be eta alpha beta minus gothic g. And with that condition, then the harmonic coordinate condition turns into a condition for the metric perturbation h which looks like so. And we call that a gauge condition. So, or we call that the harmonic gauge condition. So these are sort of the tools that we're going to use. What we're gonna do is we're gonna take the Einstein equations and we're gonna plug in our decomposition for the metric over here. But we're gonna rewrite everything in terms of the gothic metric and then decompose it like this and then expand and see what we get. Interestingly enough, this is not quite what you would do when you do linearized gravity. So pretty much everyone here took GR1 and when you take GR1, it's almost compulsory to do something in the weak field limit. And so there you just say, oh, my metric g mu nu is eta mu nu plus h mu nu. And you just go and then you derive some equations for the h mu nu and then you find that the trace reversed version of h mu nu satisfies some sort of wave equation and that's how you direct gravitational waves and blah, blah, blah. That's not how we do it in post-Newtonian theory. So to work with this h mu nu, this h mu nu is not the h mu nu that you're used to. It's similar at linear order, but it's not the same. So let's now think about how to connect those two things. I think I did it over here. Yeah, so I can invert this expression over here. You can write g mu nu at least to first order, root minus g of eta mu nu plus h mu nu. Now g or minus g is just one plus h where h is defined to be h mu nu eta mu nu. So the trace with respect to the Minkowski metric. So I can plug that over here and then I can linearize. And what I end up getting is that g mu nu is equal to eta mu nu plus h mu nu minus one half h eta mu nu plus order h square. And then you recognize this object here as h bar mu nu, the trace reverse metric perturbation. So the h mu nu that we use in post-Newtonian theory, it's like the trace reverse metric perturbation that you would use in first grade GR, but not first grade GR, the first GR semester, okay? So that's sort of the mapping between those two things. So now let's, now that we've introduced a little bit of notation, let's begin like we should always begin with Landau. So let me introduce the Landau-Lipschitz formulation of general relativity. And you will find that this Landau-Lipschitz formulation is actually quite, so I should have said that topic three here is comparable mass binaries, so comparable. So the first thing I wanna do is introduce a Landau-Lipschitz formulation. So I've already introduced the gothic metric, which is a tensor density, by the way. And there's this quantity, this awesome quantity called the h tensor. Really like this quantity. So this quantity is defined, so it's a four rank tensor. Okay, so it's defined to be two times two copies of the gothic metric, alpha, beta, new mu, anti-symmetrized on these indices. So anti-symmetrized means you take that thing and then you subtract from it the same thing, but with the indices flipped, new and beta. I think it's already been introduced in the school. And I like this because it turns out that this thing has the same symmetries in the sense of permutations. So permutation symmetries as the Riemann tensor. Wow, that's pretty cool. But what's even cooler, and I don't know how they did this in the whatever, 50s, 40s, 60s, whenever these people were alive and doing the revisions, the second derivative, d mu nu. So d mu nu here just means d mu d nu, okay? Acting on this H tensor is equal identically and without making any type of approximation to two times minus the determinant of the metric times the Einstein tensor plus 16 pi. Well, I guess it's not too surprising that you get this. T, L, L, alpha, beta. So what I am saying is that I can construct the Einstein tensor from just this very simple product of gothic metrics or tensor densities, taking two partial derivatives, no covariant derivatives, I get ta-da, I get the Einstein tensor, but not so fast. I don't get the Einstein tensor exactly. I also get this other thing called t of alpha beta L, yeah. No, no, this is in general. I think, I think it's in general. Cause we haven't done, if that's for me, just tell them I'm lecturing and so maybe for next time. So t L, L, alpha, beta, it's called the Landau-Lischt pseudotensor, a pseudotensor because it's a density. And you know, I've written a lot of horrible things in my last two lectures, especially in the first one, I like wrote the full parametric in its gory detail. I, this is where I draw the line. I am not going to write down the full Landau-Lischt pseudotensor. I'm just gonna tell you that this thing depends on the derivatives, first derivatives of the gothic metric. For example, there's this term plus like another 30 terms or so, all of which depend on the square of first derivatives of the gothic metric. So in some sense, we've separated all of the second derivatives of the gothic metric onto this term. And then we've compensated to adjust things by subtracting these extra terms that depend on first derivatives. So yeah, so now I'm pretty sure this is general. Because now what I can do is I can take that my gothic g alpha beta is eta plus or minus h alpha beta, then I can linearize in h alpha beta because any derivative of Minkowski is gonna vanish because we're working in the putter column, the non-stupid coordinates where the metric Minkowski is just one minus one or minus one, one, one, one. So these things, they just become derivatives of h. And then whenever I have a divergence of that, then I can impose the harmonic gauge and like kill a bunch of terms. But this is the formulation of gr. And something else that's sort of cool, yes, I'm gonna write over here, is that, well, obviously, since this is true, I can now use the Einstein equations to re-express g alpha beta in terms of the stress energy tensor. And so the Einstein equations that you know I love, g mu nu equal eight pi t mu nu become d mu nu, so part of the derivatives on this tensor, alpha mu beta nu equals 16 pi minus g times t alpha beta plus lambda lift-shifts alpha beta. Those are your Einstein equations now, okay? So we've isolated the second derivatives on the left-hand side, then on the right-hand side, we have the matter fields, and then we have non-linear terms that depend on squares of first derivatives of the metric. And what's sort of cool is that this thing is totally anti-symmetric on any three of its indices, just like the Riemann tensor. So I can take another partial derivative of this, the entire equation, and the partial derivative, if I take another partial derivative of this thing, so d gamma mu nu, or sorry, let me call it d alpha, mu nu of h alpha mu beta nu, this will be identically zero because the partial derivatives commute, and that's a perfectly symmetric type object, and it's acting on a quantity that's completely anti-symmetric on three of its indices, which then implies that d alpha acting on this thing on minus g times t alpha beta plus t alpha beta lambda lift-shifts has to be zero, which is like your Bianchi identity. I just written in terms of partial derivatives, yes. No. I haven't imposed a gauge condition yet. That's the same thing that that guy asked. What's your name again? I heard Gollum, but that's going to be right. Okay, so now that we have that, oh, so this is how you would derive conservation equations or flux type equations, which I'm not gonna do right now, but whatever, it's fine. Okay, so now, now let me begin to expand in H, right? Now let me make this assumption that, so 3.2, what I'm gonna show you now is the relaxed Einstein equations. Okay, so condition A, we're gonna use harmonic coordinates, meaning box on X is gonna be zero. I'm also going to use, in addition to that, even though this is not said, I'm gonna use the good coordinates, the one where the Minkowski metric is minus one, one, one, one, and this, as I said earlier, will lead to the condition d beta acting on Gothic g alpha beta has to vanish. Then the second thing I'm gonna use is the, I'm gonna introduce a metric perturbation. That's how I relax things. Okay, so again, I'm gonna use that H alpha beta equal eta alpha beta minus Gothic g alpha beta. And so this condition, these two, then become just d beta on H alpha beta has to be zero. And I'm going to now use that condition. And if I do that, then the second derivative of this H monstrosity is equal to minus box sub eta of H alpha beta plus H mu nu d mu nu acting on H alpha beta minus d mu H alpha beta. Alpha nu d nu H beta nu, nu, sorry. This is exact in that I have not perturbed in H yet. What I mean by it's exact. And now I'm gonna call this thing here. I'm gonna give it another name. I'm gonna call it 16 pi times the metric determinant times a quantity, I'm calling tau alpha beta sub H for homogeneous part, what this thing is. With that definition, my Einstein equations become the following. We have box, oh, I didn't say this. Box eta is supposed to be eta mu nu d mu d nu with partial derivatives. Just like a standard Dallam version in flat space. Yes, this H tensor? Okay, so what I've done here is I've taken this H tensor and I've taken two derivatives of this H tensor like at this level. Okay, and then I have replaced for G alpha beta, H alpha beta minus H alpha beta, taking the partial derivative, the two partial derivatives and you get this expression. So what I'm now gonna do is I'm gonna move this, so I'm gonna set this equal to that and I'm gonna move the box operator to the left hand side and write down the field equations, okay? So the field equations then become box on eta of H alpha beta equal minus 16 pi tau alpha beta, alpha beta is defined as, what is it defined as? It's defined as minus G T alpha beta plus T lambda lift sheets alpha beta plus T homogeneous alpha beta. Ta-da, so these are called the relaxed Einstein equations and they're called relaxed because I've isolated the box operator on the left hand side and I'm gonna move everything else to the right hand side and everything else to the right hand side that I have moved is not now just a nonlinear terms that depend on the square of first derivatives which is what's contained here. There's also some second derivatives that come from the homogeneous terms that appear here but I've done this because we are obsessed in physics with the green function for the flat space, the epsilon version because we know how to invert that, okay? Of course, if you're going to take everything into account, you cannot just simply invert it because on the right hand side, you don't just have matter fields or like a source, you also have H. So now if you want to solve, you do have to use perturbation theory and you need to begin to drop terms. But up to here, this is exact. Exact in quotation marks. Yeah, it's still exact, I haven't dropped anything. Okay, good. Are there questions up to there? Have you seen this before? It's pretty cool, I don't know. Good, great. Oh, and one other thing since I have this equation and I can now, right? I can now, what can I do? What would you like to do to this equation? It's an obvious thing, one more thing to do. Also on one of the boards, I'm not gonna tell you which one. Although I'm indicating directionality with my hand. No one knows what to do with this equation. There's one more equation you can derive from this equation. More or less something like that. So what if I take a partial derivative of this thing? If I take a partial derivative of this entire equation, since these are partials and partials commute and this with respect to Minkowski, I can take the partial inside. But the partial inside is zero because of the gauge condition, which then means that the partial on tau has to vanish. So as a corollary of this equation, if you want, the alpha of tau alpha beta has to be zero, which is in post Newtonian, we sometimes call this the Bianchi identity in a huge abuse of notation. Okay, now you know, yes, G. No, just G, because I want to move this minus G. No, I think it's plus G. Yeah, but there's an extra factor of minus G here and there's a minus sign here. I think they cancel. Also, you know, plus minus. I mean, if Pyotr is allowed to put plus minus, it's allowed to put plus minus. No, but I think this is right. What was I saying? Oh, yeah. Right, so we have this equation or we have this equation, right? So in GR, if you're doing fluid mechanics, have you done fluid mechanics in relativity? No, okay. So the questions of motion, how do you derive the questions of motion of a fluid in GR? What you do is you take your stress energy tensor and you take the covariant divergence of that stress energy tensor and that you can show gives you the equation of motion. So in the same spirit, this equation here tells you how the matter sources are going to move. But not just that, right? Because this thing here, this tau, contains things that depend on H. So in some sense, this equation is also telling you how the gravitational field back reacts on the motion of the particles or the point particles. Of course, these are coupled in some ways. You have, you wanna solve this, you're gonna invert it with a Green's function. These things are gonna depend on the position and the trajectories of the point particles or the war lines if you wanna call them that. But you don't know what those are until you solve this equation. So, okay. So now, let me tell you how we solve this equation. In order to solve this equation, as I said earlier, I anticipated earlier we're gonna use Green's functions. So a formal solution is that h alpha beta, just minus 16 pi box eta to the minus one acting on tau alpha beta. And so we write this down as four times the integral of a Green's function. I'm gonna call g, which has arguments x and x prime acting on tau alpha beta with argument x prime of d four x prime. And so the arguments here are supposed to be four vectors, blah, blah, blah, right? Yes, very good point. So, if tau alpha beta didn't depend on h, this is how you would solve it. Yes? Okay, so now, let's look at what tau alpha beta is here. Tau alpha beta, I was gonna say this in a little bit, you can say no. Tau alpha beta depends on the stress energy tensor of your objects plus this Landau-Liffchitz term and this h term. Let's look at the Landau-Liffchitz term. So over here, this is just further to the realty of h times further to the realty of h. So that's h square. It's of order h square. Well, times two derivatives, but still h square. Over here, we have the homogeneous piece, which is this last term in tau alpha beta. It again depends on h times two derivatives of h. So again, h square. On the left-hand side, I have something that just depends on h. So this is linear in h. So what we're gonna do is what's sometimes called a bootstrapping scheme. We're going to expand and solve this equation order by order in h. We're gonna assume that h to leading order is proportional to capital G. And then it has a correction of order g square and another correction of order g cube and so on and so forth. And you're gonna use capital G as your Protervative Expansion Parameter. And so if you do that, you'll find that these terms here are of order g square, like Newton g square. And the same as those terms over there, which means we can drop them. And then the tau alpha beta just depends on the stress energy tensor of the matter sources that you have, which you then have to prescribe. You have to have some sort of description for your matter sources as well. Well, the stress energy tensor may depend on g, not on h. So in some sense, it depends on h implicitly. But again, if your stress energy tensor depends on g, you would do an expansion about Minkowski and you would just keep the leading order part of that. That's correct. So I will get to that in a second. Yeah, so in order to solve equations like that, typically the boundary conditions are prescribed plus you're assuming that you have no incoming radiation from infinity. And so the radiation is purely outgoing. But let me continue because this will come into play in a second and if I don't answer your question by the end of the lecture, you can ask me again then. So if tau doesn't depend on h, then you get something that looks like this. And so this g here satisfies the usual equation that you would expect. Box equals to minus four pi times a direct delta of x minus x prime. So the Green's function clearly is just delta of t minus t prime minus x minus x prime divided by x minus x prime. Okay, so now let's apply our formal expansion and I'm gonna show you what the general procedure is just of these equations. So you start, this is a general procedure. So you start by expanding h to x minus x prime in powers of g. So what you have is something like h alpha beta equal capital G times k one alpha beta plus capital G square times k two alpha beta plus dot dot dot up to the nth term, j g to the n k n alpha beta. And then there's remainders of order g to the n plus one. One of these super formal, okay? So this expansion in powers of g is supposed to be understood as a Taylor series essentially about weak gravitational fields. So when we say expanding powers of g, what we mean is we expand really an expansion in weak gravity. So if your gravitational field is, for example, described by some Newtonian potential phi that is equal to g times maybe some mass divided by c square times r, then this quantity, which is dimensionless, is assumed to be much less than one, okay? That's what we mean by an expansion in g. You always have to multiply the capital G by the right factors that enter your problem to produce a dimensionless number. But this notation has become standard so we're gonna continue using it. So you expand in powers of g and then you insert that in this equation and then you solve this equation order by order in powers of g. So to zero of order, you have that k zero alpha beta which I didn't even write down is zero and that g alpha beta is just Minkowski. That's pretty much it. Like you don't even have to solve this equation. Tau, if you want to go to next order, you're gonna need to know what tau is. So tau alpha beta at zero of order is equal to this matter stress energy tensor, alpha beta, evaluated on the metric which is going to be, let's say, eta mu nu in this case. And so once I have that the metric at zero of order is Minkowski, I can go and I can calculate what tau alpha beta to zero of order is. Clearly tau l and tau h vanish and then this t which is in principle a function of g becomes the matter stress energy tensor evaluated on Minkowski. Now, if I want to go to next order, what I hit here in some sense, there's a capital G floating around. If I want to go to first order, the equation I need to now solve is box of k one alpha beta has to be equal to minus 16 pi times tau zero alpha beta evaluated on eta which again is just minus 16 pi of the matter stress energy tensor evaluated on Minkowski. And this is box acting on eta. So I've canceled the capital G that would appear on the left-hand side and on the right-hand side here. So I'm working directly with k, yes. This one, which G are you talking about? This one, this one, this one. Oh, this one, that just comes from the answer anyway. So I, in a somewhat confused derivation, I had set G's and C's to one for like a good solid chunk of the first half of this lecture. So if you look here when you substitute in for T alpha beta here, there's a capital factor of G that didn't write down. Not in front of TLL, that's correct. But TLL is of order H square and H zero is zero because the background is Minkowski. So it just vanishes. So if you want to be more precise, put the G in here and then make you happier. All right, so we are here now on this equation and this equation is essentially what you solved in first semester year. Okay, it's box of trace reverse metric perturbation at order G equal to some source which we typically put in some sort of like point particle stress energy tensor. Evaluate it on Minkowski. Yeah, I know there's three questions. Which you can then solve through the inversion of this done inversion in terms of green functions with a suitable choice of boundary conditions which I'm going to describe in a second. But before I do that, there are two questions. Yes, good. So the question is, is this valid for asymptotically flat spaces only or can one extend it to the Cedar for example or ADS? Everything I'm talking about here is for asymptotically flat space times. I don't know of any post-Newtonian development that has been done for anything but asymptotically flat with the only exception of maybe the work of Ash Tkarambonga who started developing PN theory with the Cedar boundary conditions. I think the Cedar, yeah, with like a small positive cosmological cost. Was there another question? Yeah. What I mean, what I mean is that typically the stress energy tensor, well it depends on how you treat the matter stress energy tensor, right? So the stress energy tensor has been treated differently by different relativity groups around the world. So Luc Blanchet and Thibaut D'Amour, when they, and Guillaume Fay, when they developed the post-Newtonian approximation, they model the matter sources by point particles. So there's some M and then there's integral square root of minus G times a direct delta function in detail, like I showed in the previous lecture, I believe when I was talking about hemorrhage. And then that introduces some issues with divergences that arise at higher post-Newtonian order that need to be regularized. So when I said, okay, so that's one description in terms of direct delta functions. Another description is in terms of fluid spheres. So that's the work of Clifford Will and Allen Weisman and others. So what they do there is they say, okay, stress energy tensor is, let's say, a perfect fluid, okay? And you know the stress energy tensor for a perfect fluid, it's rho U mu U nu or rho plus P U mu nu minus P G mu nu, okay? And what they do is they say, okay, each of these objects, because we're not binary, each of them is a binary, it's a perfect fluid ball with some radius, okay? Where the radius in principle is some sort of arbitrary parameter. And then they do the entire calculation. They have to do a matching calculation, but at the end of the day, at the end of the calculation, all of the terms that depend on the radius of the object better go away, or otherwise your expression depends on the finite size or the finite extent of your object. Okay, if it doesn't depend on the finite extent of your object, you can effectively then take the radius to zero and it's like you have a direct delta function. And that sense is that the two approaches become equivalent. Okay, so now to answer your question, in both of these descriptions, the stress energy tensor depends on the metric tensor, right? So when I said you take the stress energy tensor, the T mu nu matter, which is a functional of g mu nu, you evaluate it on eta mu nu. What I meant is that every time you see a g mu nu, a metric tensor entering into your description of the stress energy, you replace that g mu nu and that metric tensor by the Minkowski metric. Sorry, that was a very long-winded response. That was a complicated question. You guys ask really hard questions. Yes. And I'm gonna delay the answer to your question again because I think I'm going to talk about that, okay? Okay, so that's what you do to first order and then you can solve this through Green's function. So it's a Green's function approach here. And then that in principle gives you K1 alpha beta. And from K1 alpha beta, you can get now the metric perturbation, g, well, the full metric, g alpha beta, which is now eta alpha beta minus K1 alpha beta. And from that metric perturbation, you can now evaluate the right-hand side again, so this tau alpha beta. It's now evaluated on this g1. What I mean again is that it's a functional of the metric, so you replace the appearance of the metric by this g1 mu nu that you just solved for and you iterate. You just go around and around to higher and higher order. And if at any point you need to know what the motion of these objects is, what you do is you write down an equation for the divergence of the tau alpha beta. So, world line, world z, say alpha of tau, you can obtain this thing by looking at the equation, say tau beta, d beta of tau alpha beta, at some pn order. So let's say g mu nu evaluated at the n minus one one-th order, you set that to zero. And what that gives you is an equation that sort of looks like this, d two of tau for the eighth body in your binary, d tau squared, it's equal to something of order g plus something of order g squared plus dot dot dot something of order g to the n minus one. This case, which is a very complicated differential equation, it's essentially Newton's equation of motion, but with higher in g corrections to that, induced by the modifications of the, the modifications introduced by the curvature of space time induced by both of these objects on Minkowski space. And you solve this, you get z of tau as a series in n and in M of not just powers of g, but also powers of c. So you do a slow motion expansion. So it is really this bivariate expansion in weak fields and in slow velocities that defines what we mean by the post-international approximation. So when we say p n, what we mean is an expansion in both weak fields and slow velocities. Now, in order to solve these integrals, a lot of technical details need to be worked out related to regularity of the integrants. It's essentially everything that you were asking and boundary conditions and things like that. So let me tell you what approach people typically take to deal with integrations like this for a binary system. So I'm gonna follow here what Will and Weisman call the dire approach, which stands for direct integration of the relaxed Einstein equations. So the first thing you do is, well, you consider your binary system, right? And you center the binary system, you put the origin of your coordinate system at the center of mass of the binary. So let me call it zero here. And then there's some radial direction when working in Cartesian coordinates, but just consider the radial distance from the center of mass out to spatial infinity. Let's call this infinity. So this is just like a space like hypersurface, if you want, in your manifold. Just for one instant in time. So at some fixed instant in time, there's going to be some region that is sufficiently close to the binary. Where a set of approximations are valid and a region sufficiently far away from the binary, where a different set of approximations is valid. So let me call this separation lambda characteristic, which for a binary system is gonna be roughly the wavelength of gravitational waves emitted by the system, which is roughly the orbital separation, which I'm calling b divided by the orbital velocity, which I'm calling b. So let me make here a little drawing. So I don't need that anymore. So my advisor, then oh, and when we were doing this calculation, you call this the right egg diagram. Then when we finished the paper, he took me to a breakfast place to have sunny side eggs. Quite fun. So what we're gonna do is we're gonna, we're gonna paint the two black holes here. So one black hole here, another black hole here, and then there are some separation between these two black holes or these two bodies. They don't have to be black holes. The separation is b. These objects are moving some velocity, v1 and v2. So there are some characteristic velocity associated with this orbit. So for simplicity, let's consider, let's consider a circular orbit. It's a circle orbit. And then if this is the central mass, let's say the central mass is here, there's some distance over here, let's say that corresponds to the characteristic wavelength of your binary. So I can create a circle distance. So for the purpose of this talk, I'm gonna paint the black holes yellow. And so the regime that is between this lambda and some minimum, oh, I also need to define a minimum distance, okay? So here's the horizon of the black hole, that's the white line. But let me define some other two sphere here. It's a two sphere inside of which I'm not going to evaluate fields because if I try to evaluate fields, everything diverges violently. So I'm gonna cut that out from my domain and I'm only gonna look for solutions that are outside of that domain. So this is Terra incognita for post-Newtonian theory. And what this radius is, it can be a few m or 10, it's some number of m away from the horizon. Exactly what that is, it's not known because the series, the purpose of the series are not entirely clear. But what you can do is you can compare your post-Newtonian solution to numerical relativity and ask up to how close can I evaluate my metric and relative to my, when does the post-Newtonian approximation stop being accurate relative to a few numerical simulations and you find that the distance between the horizon and this radius that I painted here is a few m, depending a little bit on the system, right? So then the regime that is between lambda and the origin minus the origin the purple balls, that regime, we're gonna call the near zone. And the regime that is outside of lambda, we're gonna call that in a very clever choice of terminology, the far zone. Sometimes people call this the radiation zone, sometimes they call it the wave zone, whatever, near zone, far zone. And this is why they, you know, he took me out to eat some fried eggs because it sort of looks like two eggs, yeah. Two eggs, like you break them together and then you put them on a bed of french fries, it's like favorite food, right there. Anyway, what was I saying? Oh, right, so you divide this region and so you're gonna employ, so let me just put in some characteristic distance here. So what we're gonna do is, what did I paint, blue and green, okay? So blue and green, so what we're gonna do is we're gonna use some approximations from here to here, roughly, this is the near zone. And then we're gonna use a different set of approximations, like from here all the way to special infinity. I'm gonna call that the far zone, okay? And then what we're gonna find is that because everything here is asymptotic, there's a regime roughly between here and here, buffer zone, where I can expand both approximations. I can expand the far zone approximation about this edge and I can expand the near zone approximation about this edge and those two double asymptotic expansions are gonna be asymptotically equivalent in this buffer zone region, which by the space surface is a shell, right? You're supposed to imagine rotating this, okay? So in this shell, space like shell, a spatial shell, then the two approximations need to work. So when you impose boundary conditions, you impose boundary conditions on sort of the far zone metric. So you impose no incoming radiation conditions, say here. And here you impose conditions such that this metric, asymptotically matches that metric in this region, okay? And then here, you have to be careful about the realization of certain terms because when you treat the stress energy tensor as point particles, then at some post-Newtonian order, you're inevitably going to run into integrals that are highly divergent, okay? Like integrals, DR, say, of one over R to the four or one over R to the five or to high power. And if you formally extend like you do in the post-Newtonian treatment of the French group, these integrals all the way to zero, then you have singularities here. So in order to remove the singularities, you can use things like Hadamard-Partifini or you can use dimensional regularization. It's what people are using nowadays. That's where you do these integrals in D dimensions and then you take the limit as D goes to three. So I am, that's no way I can discuss any of that, but it's just not of that. All right, so how much more time do I have? I lost track. 10 minutes, right? 10 minutes, okay, I'm gonna go a little bit longer, okay. Good, so what's sort of nice about this is that remember these integrations that I sketched over here, okay? They're over the past light cone about any, no, field point area, right? So if you wanna calculate the behavior of the field, that particular field point in space time, then what you do is you need to integrate this on the past light cone, about that point. That's a retarded Green's function, right? And so if I'm gonna go and split my domain into a far zone and a near zone, essentially I'm gonna have an integration over a subset of this past light cone and then another integration over the rest of the light cone, okay? But if I am very close to the binary, like within a gravitational wavelength, then one of the nice approximations that you can make is the following, making a mess. So the near zone is defined to be the regime where r is much less than this characteristic wavelength. The far zone is defined to be the place where r is much larger than this characteristic wavelength. Okay, and then if you look at the retarded time, which is gonna look like, you know, it'll be something like t minus r, like that, then clear that tau r is gonna be much less than lambda c in the near zone and tau r is gonna be, in principle, much larger than lambda c in the far zone. So whenever you have a field, let's call it a, and you calculate r times the partial derivative of this field, a dt. If a, which is typically a wave, is varying on a length scale of lambda, then this is roughly r a over lambda c. But this, because there is an r over lambda c and in the near zone, r over lambda c is much less than one. This is much less than a in the near zone. So from that expression, you can derive that dA dt divided by dA, d any spatial coordinate, I'm gonna call it xi, must be much less than one in the near zone. That is to say, retardation effects in the near zone are subdominant. Retardation effects cannot be ignored in the far zone. Okay, so if you have a quantity in the near zone that depends on retarded time, then you can expand that quantity, if you have t minus r, you can expand it about t. And so it becomes, a quantity that depends on t minus r becomes a quantity that depends on t plus derivatives. If you do it's Taylor expansion, f of t minus r becomes f of t plus f dot of t times r plus dot, dot, dot, things like that, schematically speaking, okay. And so this is a series of approximations that you can make in the near zone. You're essentially taking, if you wanna think geometrically, this integral that was supposed to be over the entire past light cone and you're considering the part of this integral, so the part of this light cone that is in the near zone and it's supposed to be this sort of inclined subsurface and then you're just essentially representing that as a Taylor expansion about t is equal to zero. So you're bending that surface back to be normal plus corrections as you go. So those are techniques, for example, that you use a lot when you invert this delambersion and you have this x minus x prime in the denominator and you sort of do progressive expansions. In the far zone, it turns out you can do a formal multipolar expansion in parts of G and get the exact solution exactly. So then at the end of the day, once you've solved for the fields in the near zone and in the far zone, you sort of reconstruct everything. You reconstruct the metric perturbation and so you write h mu nu is equal to say h mu nu in the near zone plus your h mu nu in the far zone and you work with that quantity. You calculate observables with that quantity and so on and so forth, okay? So just to give you an example, once you have this metric perturbation, you can construct the plus polarization of the gravitational wave, which is essentially some projection I'm gonna call p plus jk acting on the metric perturbation hjk. And what you get for a binary system is something that looks like two times a quantity called eta times the total mass times dl times a quantity that we call x times things that look like h naught plus h one half x one half plus dot dot dot. And here eta is called a symmetric mass ratio, which is just m one times m two divided by the total mass squared. Then m is the total mass, just m one plus m two. dl is a luminosity distance, so it's just the distance for us. And x is a quantity called m, well that's defined to be m times omega to the two thirds. So m is the total mass and omega is the orbital angular frequency of your binary. So if you wanna, this is essentially our post Newtonian expansion parameter, this low motion expansion parameter, that's a capital factor of g here also, hiding. So why is this an expansion parameter? Because omega by Kepler's law to like leading order is say the mass divided by the separation cubed to the one half. So then m omega or x is m over b cubed, m over b cubed, cubed to the one half to the two thirds because there's a factor of m here, so you put under the square root, it becomes m square. So m squared times m is m cubed divided by b cubed, everything to the one half, and then I have conveniently put this to the two thirds, that's how I defined x. So then this quantity is roughly m over b, but m over b, that's the mass, total mass divided by the orbital separation, which by the virial theorem is roughly v square. So you see that this is really, as written here, an expansion in powers of v, where this is your controlling factor and these are the prior order corrections. And this is some functions that I didn't write down that depend on the orbital phase of the binary and things like that. And so there's this annoying thing in the literature that relates to post-Newtonian counting that you should be familiar with, and I promise like, I'll stop here because I know that everyone is still trying to digest food and beginning to like dose off. But people talk about one pn calculations and two pn calculations and three pn calculations all the time, and I sort of feel like you need to know what those are, even if you decide not to work on post-Newtonian theory ever, okay? So a term of npn order is one that scales as v over c to the 2n relative, or with respect to, it's controlling factor. The controlling factor in this case is this x, right? It's the thing that you pull out at what we would call leading order in your series, okay? So this goes as m over dl, so it has a power of g here, and then there's a v squared over c square, okay? But this term that's proportional to v, so this goes as v square, this goes as v, okay? So if I just keep on going with a series, there's gonna be an h1 that goes as x plus dot dot dot, okay? So this would go as v squared, right? This term that goes as v squared is called one pn order, even though there's an x over here, and in absolute ordering, this times that is x squared, which goes like v to the four, which might lead, it always leads to you think that that's two pn order. It's not two pn order, it's one pn order. This is half a pn order. This is zero pn order, also known as Newtonian, but that makes no sense. You're gonna say, because that gravitational wave, there are no gravitational waves in Newtonian theory. Do not blame me for the nomenclature, okay? But that's how it's done, and the controlling factor changes from quantity to quantity. So for example, if you calculate the rate of change or the binding energy of your binary system, e dot, which is also the power emitted in gravitational waves, if you work in this long enough, you know that this is 32 over five times eight a square times v to the 10 times a zero is that looks like one plus something here times v square plus something here times v cubed plus dot dot dot. Okay, so that's the amount of energy that's taken or carried by gravitational waves. You probably calculated this in your first semester class. It is the squared of the third derivative of the quadruple moment. So I triple dot squared. Okay, that's what you do if you calculate, this is what you get if you calculate that. This term goes as v to the 10. So you might think, oh, this is 5 pn order. This is Newtonian, because that's a controlling factor. So this term one is Newtonian. This is 1 pn. This is the 1.5 pn correction and so on and so forth. Okay, even though this v to the 10 is different from this v square and this quantity. And I think probably gonna stop there. I just wanna mention before we go that a big elephant in the room is that I'm supposed to be talking about black holes and I've sort of excised from my domain the black hole. So the lecture tomorrow is gonna be about how do we deal with that region, the black hole region. Okay, thank you.